Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on (...) results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results. (shrink)

An Exact Pair for the Arithmetic Degrees whose join is not a Weak Uniform Upper Bound, in the Recursive Function Theory-Newsletters, No. 28, August-September 1982.